Help with "y=ax^2+bx+c" in relation to parabola please.

[teetar]

I have to do a math project and I'm stuck on this one point. I've narrowed down the following for "y=ax^2+bx+c":

The "a" value will change the orientation of the parabola (i.e. a positive number will have a positive parabola and vice verse), and I've figured that if the a>1/(-1) then the parabola will be narrower, and if it's less than those numbers than the parabola will be wider.

For the "c" value, I've connected it in all cases so far to the Y-intercept of the parabola.

The "b" value of the equation (y=ax^2+bx+c) has been the most difficult one to pinpoint to a certain action in the shaping of a parabola. I at first figured it to be related to the vertex's X-coordinate, however, that data was inconsistent.

The program I have been using in order to conduct these "tests" with different values is "GeoGebra".

I'm not asking for answers, that would be cheating, but I am in need of some sort of hint.
Any help would be greatly appreciated, so thanks!

 

[pka]

I have to do a math project and I'm stuck on this one point. I've narrowed down the following for "y=ax^2+bx+c":

The "a" value will change the orientation of the parabola (i.e. a positive number will have a positive parabola and vice verse), and I've figured that if the a>1/(-1) then the parabola will be narrower, and if it's less than those numbers than the parabola will be wider.

For the "c" value, I've connected it in all cases so far to the Y-intercept of the parabola.

The "b" value of the equation (y=ax^2+bx+c) has been the most difficult one to pinpoint to a certain action in the shaping of a parabola. I at first figured it to be related to the vertex's X-coordinate, however, that data was inconsistent.

The program I have been using in order to conduct these "tests" with different values is "GeoGebra".

I'm not asking for answers, that would be cheating, but I am in need of some sort of hint.
Any help would be greatly appreciated, so thanks!

Here is a suggestion: multiply out \(\displaystyle (x-k)(x-j)\).
You see \(\displaystyle x^2-(k+j)x+kj\). Now once you realize that \(\displaystyle k~\&~j\) are roots then you can relate the sum of the roots and the product of the roots to the coefficients \(\displaystyle a,~b,~\&~c\)

 

[teetar]

Here is a suggestion: multiply out \(\displaystyle (x-k)(x-j)\).
You see \(\displaystyle x^2-(k+j)x+kj\). Now once you realize that \(\displaystyle k~\&~j\) are roots then you can relate the sum of the roots and the product of the roots to the coefficients \(\displaystyle a,~b,~\&~c\)

First off, thank you very much for responding. I'm sorry, but I cannot understand what you are explaining :/. I don't know if what I am saying is stupid or not, however, I cannot grasp what you're saying, and I haven't seen this (x-k)(x-j) equation or the multiplied version. I will admit that there are many gaps in my mathematics education. When you say "roots" what do you mean by that? Again, thanks for your response, and I'm sorry that I cannot quite grasp what you are saying.

 

[pka]

First off, thank you very much for responding. I'm sorry, but I cannot understand what you are explaining :/. I don't know if what I am saying is stupid or not, however, I cannot grasp what you're saying, and I haven't seen this (x-k)(x-j) equation or the multiplied version. I will admit that there are many gaps in my mathematics education. When you say "roots" what do you mean by that? Again, thanks for your response, and I'm sorry that I cannot quite grasp what you are saying.

In the quadratic equation
\(\displaystyle ax^2+bx+c=0\) the sum of the roots is \(\displaystyle -\dfrac{b}{a}\) and the product of the roots is \(\displaystyle \dfrac{c}{a}\).

 

[JeffM]

First off, thank you very much for responding. I'm sorry, but I cannot understand what you are explaining :/. I don't know if what I am saying is stupid or not, however, I cannot grasp what you're saying, and I haven't seen this (x-k)(x-j) equation or the multiplied version. I will admit that there are many gaps in my mathematics education. When you say "roots" what do you mean by that? Again, thanks for your response, and I'm sorry that I cannot quite grasp what you are saying.

The roots of a function are the x-intercepts of the function's graph.

Every quadratic function with real coefficients can be written in the form: \(\displaystyle ax^2 + bx + c,\ where\ c \ne 0\ and\ a,\ b,\ c \in \mathbb R.\)

Every quadratic has no real roots, one distinct real root, or two distinct real roots.

The roots are given by the quadratic formula: \(\displaystyle \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\)

\(\displaystyle If\ b^2 - 4ac > 0,\ the\ quadratic\ has\ two\ distinct\ real\ roots.\)

\(\displaystyle If\ b^2 - 4ac = 0,\ the\ quadratic\ has\ one\ distinct\ real\ root.\)

\(\displaystyle If\ b^2 - 4ac < 0,\ the\ quadratic\ has\ no\ real\ roots.\)

So the value of b is ONE element in determining whether there are one or more real roots and if so where.

The value of b is also ONE element in determining the vertex. That is, your guess was in the right direction, just too simple.

I suggest doing the following to get an intuition. Graph three quadratics, leaving a and c the same in all three quadratics but changing b.

Finally, a quadratic with real coefficients can be expressed as a product of a and two linear real factors or two linear complex factors.

\(\displaystyle a\left(x - \dfrac{-b + \sqrt{b^2 - 4ac}}{2a}\right)\left(x - \dfrac{-b - \sqrt{b^2 - 4ac}}{2a}\right) =\)

\(\displaystyle a * \left(x^2 - \dfrac{x\left(- b - \sqrt{b^2 - 4ac}\right)}{2a} -\dfrac{\left(- b + \sqrt{b^2 - 4ac}\right)x}{2a} + \dfrac{- b + \sqrt{b^2 - 4ac}}{2a} * \dfrac{- b - \sqrt{b^2 - 4ac}}{2a}\right) = \)

\(\displaystyle a * \left(x^2 - \dfrac{- bx - bx - x\sqrt{b^2 - 4ac} + x\sqrt{b^2 - 4ac}}{2a} + \dfrac{(- b)^2 - \left(\sqrt{b^2 - 4ac}\right)^2}{4a^2}\right) = \)

\(\displaystyle a * \left(x^2 - \dfrac{- 2bx}{2a} + \dfrac{b^2 - (b^2 - 4ac)}{4a^2}\right) = \)

\(\displaystyle a * \left(x^2 + \dfrac{bx}{a} + \dfrac{4c}{4a}\right) = \)

\(\displaystyle ax^2 + bx + c.\)